Sending an Investment Memo As a Way to Reconnect

I am waiting to hear back re first round interview invite from some pension funds and sovereign wealth funds that I am keen on. I am currently unemployed (long story, none of is my fault or performance-related). I had initial conversations with some principals at these firms 2 months ago with whom I later shared some "well-written memos" that I had prepared independently a year or so ago so as to give these folks a better idea of my investment thought process. I recently prepared another memo (one week ago), which I know is solid and compelling. As a way to reconnect with these principals, I am thinking if I should send my new memo and tell them briefly that why I think is opportunity is exciting and if they want additional color, I'd be happy to provide. Later during the email, I can slip in a cheeky question: "have you guys given any thought to strengthening the team?" What do the pros suggest?